Ok, well I finally disconnected my device that is shown in the picture and brought it in to my room to test and look further. Strangely after I removed it from the wood backer and separated the shield, and put it all back together it seemed to work just fine all day long (in my room) (this was Friday). But even though it was working fine, I still added an LED/resistor to one of the pins on the Arduino and updated the code in the sketch to turn on and off that LED whenever it received a garage door open or close command. It all seemed to work fine (including the relay happily clicking away when it should) still in my room also with it happily turning on/off that LED whenever it received a close/open command from my Android. So I installed it back up in my garage near the opener and hooked it all up, and seemed to work all day (this was yesterday).
Then this morning it worked once, then it didn't work anymore. 
And what I dreaded may happen happened - the LED never turned on/off when I would send the open or close command. According to my 'deductive skills', this means that nothing is wrong with the relay/transistor or that pin on my Arduino, correct?. Which, in turn, means that I've got something else going on such as a problem with the Ethernet Shield maybe? Now I am very overwhelmed. I don't really know how/what to troubleshoot something as complex as that shield.
Any help/advice would be greatly appreciated...but please remember, I'm a noob, so the more detail, the better.
Thanks